In the last 5 or 10 years, I have found an increase in the difficulty
students have with place value and numeration and rounding. Is there some
strategy, exercise, or drill to get these concepts across to kids, some
manipulatives that could be used?

Can you explain why some bit combinations for 2421 are invalid? For
example, to represent decimal 5 (five) in 2421, the correct code is
1011. But why can't we represent decimal 5 in 2421 as 0101, because
if we add the weights as 0*2 + 1*4 + 0*2 + 1*1, we get 5 in decimal.

I have to estimate 32 - 15 by rounding the numbers to the nearest 10.
I know 32 goes to 30, and rounding rules say 15 should go to 20. But
that makes my estimated answer 10, when the real answer is 17. If I
round the 15 to 10, I get an estimate of 20, which is closer to 17.
Why does following the rounding rule give me a worse estimate?

When converting a negative mixed number into an improper fraction, it
seems like we ignore the rules for integer addition. For example, to
convert -4 1/7 we think -4 1/7 = -4 * 7 + 1 = -28 + 1 = -27/7. But the
correct answer is -29/7. How does -28 + 1 make -29?

On internet communities it is common to reduce figures using 'k' as a
shortcut; for example 2k is used instead of 2000. I was under the
impression that a number after the k (such as 2k4) meant 2400, and
that to get 2004 in this system you'd need to write 2k004. But I've
also seen 2k4 used for 2004. Which is correct? Does 2k4 mean 2004 or
2400?